Minimal subsystems of affine dynamics on local fields

We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field of p -adic numbers, for any non-trivial affine dynamical system, we prove that the field is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on . For each given prime p , there is a finite number of conjugacy classes.